Simulations of estimating causal effects using undersmoothed HAL
Seraphina Shi
2023-03-17
Simulation 3
Data structure: \(O = (W, A, Z, Y)\)
- U - exogenous variables
- W - baseline covariate that is a measure of body condition
- A - treatment level based on W, continuous between 0 and 5
- Z - intermediate curve based on W and A
- Y - outcome, indicator of an event ?
Underlying data generating process, \(P_{U,X}\)
- Exogenous variables:
- \(U_A \sim Normal(\mu=0, \sigma^2 = 1^2)\)
- \(U_A \sim Normal(\mu=0, \sigma^2 = 2^2)\)
- \(U_Z \sim Uniform(min = 0, max = 1)\)
- \(U_Y \sim Uniform(min = 0, max = 1)\)
- Structural equations F and endogenous variables:
- \(W = U_W\)
- \(A = bound(2 - 0.5W + U_A, min=0, max=5)\)
- \(Z = \mathbf{I}[U_Z < expit(2-W-A)]\)
- \(Y = \mathbf{I}[U_Y < expit(-10 - 3W + 2A + 10Z + Z(10 + 5W + 2AW))]\)
## W A Z Y
## Min. :-3.777088 Min. :0.000 Min. :0.0000 Min. :0.0000
## 1st Qu.:-0.646166 1st Qu.:0.584 1st Qu.:0.0000 1st Qu.:0.0000
## Median : 0.003599 Median :1.975 Median :0.0000 Median :0.0000
## Mean : 0.007583 Mean :2.100 Mean :0.4858 Mean :0.2532
## 3rd Qu.: 0.673761 3rd Qu.:3.386 3rd Qu.:1.0000 3rd Qu.:1.0000
## Max. : 3.725200 Max. :5.000 Max. :1.0000 Max. :1.0000## Summary of A given W < -1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 1.464 2.843 2.753 4.200 5.000## Summary of A given -1 < W <= 0:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.8976 2.2133 2.2891 3.6245 5.0000## Summary of A given 0 < W <= 1:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000 0.3664 1.7641 1.8995 3.0844 5.0000## Summary of A given 1 < W:## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 0.000 1.152 1.499 2.536 5.000
n = 200
CV HAL
results
## CV selected lambda (from one sample): 0.0041344192189172
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00502291469916564## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000965685169251822
## which is 0.233572146909012 * lambda_CV
n = 500
CV HAL
results
## CV selected lambda (from one sample): 0.00179663135270864
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00167069905743632## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 0.000291732819942546
## which is 0.162377673918872 * lambda_CV
1000 repetition
## The average of unsersmoothed lambda (from 1000 sample): 0.000270475505996292
## which is 0.155768441258726 * the average of 1000 lambda_CV## z=1:
## z=0:
Oevr a grid of lambda scalers
n = 1000
CV HAL
results
## CV selected lambda (from one sample): 0.000284242765357572
1000 repetition
## The average of CV selected lambdas (from 1000 sample): 0.00072537755501008## z=1:
## z=0:
Globally Undersmoothed HAL
results
## Undersmoothed lambda: 4.61546787671019e-05
## which is 0.162377673918872 * lambda_CV